M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 67
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In the illustration of Figure 9-21 thelattice positions are occupied by spheres which have the highestpossible symmetry, while there is no discernible symmetry of thelattice. However, the building elements have usually lower symmetries, especially in molecular crystals. There is a common misconception about the difference between the crystal and its lattice. A crystalis an array of units (atoms, ions, or molecules) in which a structuralmotif is repeated in three dimensions.
A lattice is an array of points,and every point has the same environment of points in the same orientation. Each crystal has an associated lattice, whose origin and basisvectors can be chosen in various ways. Accordingly, for example, itFigure 9-21. Artistic expression of atomic arrangement in an extended structure.Sculpture Cosmonergy by Kawan-Mo Chung in downtown Seoul (photograph bythe authors).9.4. The 230 Space Groups437would be improper to speak about “interpenetrating lattices,” while itis correct to talk about interpenetrating arrays of atoms [27].The system of the 230 three-dimensional space groups was established a long time ago, well before X-ray diffraction could have beenapplied to the determination of crystal structure.
These 230 threedimensional space groups were derived in their entirety by Fedorov,Schoenflies, and Barlow working independently at the end of the 19thcentury, and it will always be considered a great scientific feat. Theamateur Barlow considered oriented motifs; his method was hangingpairs of gloves on a rack to make space group models. It was a trulyempirical approach. “He bought gloves by the gross, so the story goes,mystifying the sales lady by answering ‘I don’t care’ to her first question, ‘What size sir?’ ” [28].An interesting statistical test was performed concerning the totalnumber of three-dimensional space groups some time in the mid1960s [29]. It was a uniquely appropriate point in the history ofcrystallography for such a test: Already a large number of crystalstructures had been determined, but examples of all the space groupshad not yet been found among the actual crystals.
The total number ofthree-dimensional space groups had long before been firmly established. Thus, the test was considered as much to be a check ofthe applied statistical method as to be a source of crystallographicinformation. Although there are 230 space groups, not all of themare in practice distinguishable. So 11 enantiomorphous groups wereexcluded from the count as were two more groups for other reasons.Thus, the number of space groups to be considered was 217.
The3782 crystal structures that were reviewed showed a wide variation inthe frequency of occurrence of the different space groups. One groupoccurred 355 times, while 33 groups occurred only once each. It wasalso interesting that only 178 groups out of the total of 217 occurred.Based on the available data of the distribution of the space groupsamong the determined structures, the findings were extrapolated toan indefinitely large sample. The statistical test led to an extrapolatedvalue of 216. The estimated accuracy of the procedure was 2%. Thus,the estimate agreed with the accepted value of the total number of thepractically distinguishable space groups of 217.The statistical analysis has also been applied separately to thedata on inorganic and organic crystals.
In both cases the extrapolatedestimate for the total number of three-dimensional space groups wassmaller than when all data had been considered together. The total4389 Crystalsnumbers estimated for the inorganic and organic structures were 209and 185, respectively. Thus the conclusion could be made that theinorganic and organic crystals belong to space groups with differentpopulation distributions. Statistical analysis of population distributions among the three-dimensional space groups, according to variouscriteria, has remained an important research tool [30].In the following section, we will look in some more detail at thesymmetry systems of two fundamentally important crystals, rock saltand diamond, following the descriptions by Shubnikov and Koptsik[31].
The descriptions will be far from complete; they will aimat giving some flavor for the characterization of these two highlysymmetrical structures.9.4.1. Rock Salt and DiamondThe unit cell of the rock salt structure and the projection of this structure along the edges of the unit cell onto a horizontal plane are shownin Figure 9-22a and b. The equivalent ions are related by translationsa = b = c along the edges of the cube, or (a + b)/2, (a + c)/2, (b + c)/2along the face diagonals. All this corresponds to the face-centeredcubic group (F). The structure coincides with itself not only after thesetranslations, but also after the operations of the point group m3m (orin other notation 6̃/4). The point-group symmetry elements are shownalso in Figure 9-22c.
The symmetry elements of this group intersectat the centers of all ions and thus they become symmetry elements forthe whole unit cell and, accordingly, for the whole crystal.Among the projected symmetry elements in Figure 9-22c, there aresome which are derived from the generating elements. This is thecase, for example, for vertical glide-reflection planes with elementary translations a/2 and b/2 (represented by broken lines), translations(dot-dash lines), vertical screw axes 21 and 42 , and symmetry centers(small hollow circles, some of which lie above the plane by 1/4 of theelementary translation).Two very simple descriptions of the rock salt crystal structure arealso given.
According to one, the sodium and chloride ions occupypositions with point-group symmetry m3m forming a checkeredpattern in the Fm3m space group. According to the other description,the structure consists of two cubic sublattices in parallel orientation,one of sodium ions and the other of chloride ions.9.4. The 230 Space Groups439Figure 9-22. The crystal structure of the rock salt after Shubnikov and Koptsik[32]. (a) A unit cell; (b) Projection of the structure along the edges of the unit cellonto a horizontal plane; (c) Projection of some symmetry elements of the Fm3mspace group onto the same plane.
The vertical screw axes 21 and 42 are marked bytheir respective symbols. Used with permission.Figure 9-23 illustrates the diamond structure. It can be regardedas a set of two face-centered cubic sublattices displaced relative toeach other by 1/4 of the body diagonal of the cube. Each of the twosublattices has the F43m space group, and in addition there are someoperations transforming one to the other. The complete diamond structure has the space group Fd3m, where “d ” stands for a “diamond”plane.Among the projected symmetry elements in Figure 9-23c, there areagain some which are produced by the generating elements.
Specialfor the diamond structure are the symmetry elements which connectthe two subgroups F43m. They include vertical left-handed and righthanded screw axes, 41 and 43 , respectively, symmetry centers (smallhollow circles, 1/8 and 3/8 of the elementary translation c abovethe plane), vertical “diamond” glide-reflection planes d representedby dot-dash lines with arrows, and similar systems of connectingelements in the horizontal directions.4409 CrystalsFigure 9-23. The diamond structure after Shubnikov and Koptsik [33]. (a) A unitcell; the edges of the cube are the a, b, and c axes; (b) Two face-centered cubicsublattices displaced along the body diagonal of the cube; (c) Projection of somesymmetry elements of the Fd3m space group onto a horizontal plane.
The verticalscrew axes 41 and 43 are marked by appropriate symbols. Used with permission.The subgroup F43m is common both to the rock salt space groupFm3m and to the diamond space group Fd3m. The space groupFd3m is obtained from Fm3m by replacing the symmetry planes mby glide-reflection planes d with the latter displaced 1/8 along thecube edges.9.5. Dense PackingDalton [34] envisaged the structural difference between water and icein packing properties. Figure 9-24 reproduces a drawing from his 1808book A New System of Chemical Philosophy. According to Dalton, the“atoms” of ice arrange themselves in a hexagonal scheme, while the“atoms” of water do not.
In any case it is remarkable that the principal difference between the water and ice structures is expressed inpacking density. Figure 9-25 originates from a different age and showsthe atomic and molecular arrangements in the crystals of 2Zn-insulinfrom the work of Dorothy Hodgkin and her associates [35]. The9.5. Dense Packing441Figure 9-24. Dalton’s models for water (1 and 3) and ice (2 and 4–6) [36].molecular structure of insulin is extremely complicated but the molecular packing, especially the arrangement of the insulin hexamers,reminds us of Dalton’s hexagonal ice.The symmetry of the crystal structure is a direct consequence ofdense packing. The densest packing is when each building elementmakes the maximum number of contacts in the structure. First,the packing of equal spheres in atomic and ionic systems will bediscussed.
Then molecular packing will be considered. Only characteristic features and examples will be dealt with here, followingKitaigorodskii [38].∗∗A. I. Kitaigorodskii’s name appears in two versions in this book and even morein the literature. The rules of transliterations from the original Russian suggestKitaigorodskii, but his name was transliterated on many of his publications asKitaigorodsky.4429 CrystalsFigure 9-25. Atomic arrangement in the 2Zn-insulin crystal. The smaller projection drawing shows the molecular packing in the insulin hexameters.
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